properties of knots preserved by cabling -...

communications inanalysis and geometryVolume 19, Number 3, 541–562, 2011

Properties of knots preserved by cabling

Alexander Zupan

We examine geometric properties of a knot J that are unchangedby taking a (p, q)-cable K of J . Specifically, we show that w(K) =q2 · w(J), where w(K) is the width of K in the sense of Gabai. Weuse this fact to demonstrate that thin position is a minimal bridgeposition of J if and only if the same is true for K, and more gener-ally we show that an “obvious” cabling of any thin position of J isa thin position of K. We conclude by proving that J is meridionallysmall (mp-small) if and only if K is meridionally small (mp-small),and if J is mp-small and every non-minimal bridge position for Jis stabilized, then the same is true for K.

1. Introduction

In [14], Horst Schubert introduces the bridge number b(K) of a knot K inS3 and proves that for a satellite knot K with companion J and patternK with wrapping number n, b(K) ≥ n · b(J). This is also proven by Schul-tens [15]. In particular, if K is a (p, q)-cable of J , then Schubert’s boundis sharp; that is, b(K) = q · b(J). Bridge number can be approached as anelement of the broad collection of topological invariants that can be cal-culated by minimizing some sort of numerical complexity over all possiblepositions of a space with respect to a Morse function. Knot width, intro-duced by Gabai [5], also falls into this category, as do Heegaard genus andthin position of three-manifolds. In fact, knot width can be seen as a typeof loose refinement of bridge number: for any knot K whose exterior con-tains no essential meridional planar surfaces, w(K) = 2 b(K)2 and any thinposition is a minimal bridge position for K [17]. We call any knot satisfyingw(K) = 2 b(K)2 bridge-thin and denote the collection of all such knots BT .For knots K not in BT , the situation becomes more complicated: recentlyBlair and Tomova have shown that the bridge number of a knot K cannotalways be recovered from the thin position of K [2].

In order to further develop the connection between bridge number andknot width, we make the following conjecture in [19] as an analogue to theabove theorem of Schubert and Schultens:

541

542 Alexander Zupan

Conjecture 1.1. If K is a satellite knot with companion J and pattern Kwith wrapping number n, then

w(K) ≥ n2 · w(J).

One purpose of this paper is to prove the conjecture for cable knots andto show that the bound is sharp, as with bridge number. We prove

Theorem 1.1. If K is a (p, q)-cable of J , then

w(K) = q2 · w(J),

and an “obvious” (p, q)-cabling of any thin position of J is a thin positionof K.

In particular, this shows that J is in BT if and only if some cable of Jis in BT .

The collection BT contains many knots. Let MP denote the collectionof knots K whose exterior E(K) contains no essential meridional planarsurface (mp-small knots), let M denote the collection whose exterior E(K)contains no essential meridional surface (meridionally small knots), and letS consist of those knots whose exterior E(K) contains no essential closedsurface (small knots). By Culler-Gordon-Luecke-Shalen [4], S ⊂ M , clearlyM ⊂ MP, and by Thompson [17], MP ⊂ BT . In summary,

S ⊂ M ⊂ MP ⊂ BT .

It is well known that S �= M ; for instance, see Proposition 1.6 of [10].Examples of knots K ∈ BT such that K /∈ MP appear in [1, 6]. We arenot aware of examples of knots K in MP and not in M , but we suspectthat such examples exist.

For any knot J and (p, q)-cable K of J , we have that K /∈ S as the com-panion torus is essential in E(K), so smallness is not preserved by cabling.However, since cabling does preserve containment in BT , we can examinethe situation when J ∈ M or J ∈ MP. We prove

Theorem 1.2. For a knot J , the following are equivalent:

(a) J ∈ M .

(b) Every cable K of J is in M .

(c) There exists a cable K of J in M .

Properties of knots preserved by cabling 543

The same statement is true if MP replaces M .

Thus, meridional smallness and mp-smallness are also preserved bycabling.

Finally, a natural problem concerning bridge number is to determinefor which knots are all non-minimal bridge positions stabilized. A bridgeposition is stabilized if a minimum and maximum can be cancelled to createa presentation with a lower bridge number. In this direction, Otal showsin [11] that every non-minimal bridge position of the unknot or a two-bridgeknot is stabilized, and Ozawa has recently proved the same statement fortorus knots in [12]. We follow Ozawa’s proof in order to show the following:

Theorem 1.3. Suppose K is a (p, q)-cable of J , where J is mp-small.

(a) If every non-minimal bridge position of J is stabilized, then every non-minimal bridge position of K is stabilized.

(b) The cardinality of the collection of minimal bridge positions of K doesnot exceed the cardinality of the collection of minimal bridge positionsof J .

2. Preliminaries

Fix a Morse function h : S3 → R such that h has exactly two critical points,which we denote ±∞. Now, fix a knot K and let K denote the set of allembeddings k of S1 into S3 isotopic to K and such that h|k is Morse. Foreach such k, there are critical values c0 < · · · < cp of h|k. Choose regularvalues c0 < r1 < · · · < rp < cp of h|k, and for each i, let xi = |k ∩ h−1(ri)|;thus, we associate the tuple (x1, . . . , xp) of even integers to k. Define thewidth of k to be

w(k) =∑

xi

and the bridge number of k, b(k), to be the number of maxima (or minima)of h|k. Then the width and bridge number of the knot K are defined as

w(K) = mink∈K

w(k) and b(K) = mink∈K

b(k).

Any k ∈ K satisfying w(k) = w(K) is called a thin position of K, and anyk with all maxima above all minima is called a bridge position of K. If k is abridge position and b(k) = b(K), we call k a minimal bridge position of K.

For any k ∈ K with associated tuple (x1, . . . , xp), we say xi correspondsto a thick level if xi > xi−1, xi+1, and xi corresponds to a thin level if

544 Alexander Zupan

xi < xi−1, xi+1, where 2 ≤ i ≤ n − 1. Thus, we can associate a thick/thintuple of integers (a1, b1, a2, . . . , bn−1, an) to k, where each ai corresponds toa thick level and each bi corresponds to a thin level. We also have a collectionof level surfaces Ai and Bi satisfying |k ∩ Ai| = ai and |k ∩ Bi| = bi. We callthese surfaces thick and thin surfaces, respectively, for k. From [13],

(2.1) w(k) =12

(∑a2

i −∑

b2i

).

The definitions in the next two paragraphs are taken from [16]. Given anembedding k, there may exist certain isotopies that decrease w(k), as deter-mined by the intersections of strict upper and lower disks. An upper disk fork at a thick surface Ai is an embedded disk D such that ∂D consists of twoarcs, one arc in k and one arc in Ai, where the arc in k does not intersectany thin surfaces and contains exactly one maximum. A strict upper disk isan upper disk whose interior contains no critical points with respect to h. Alower disk and strict lower disk are defined similarly.

If there is a pair (D, E) of strict upper and lower disks for k at Ai thatintersect in a single point contained in k, we can cancel out the maximumand minimum of k contained in D and E, and w(k) decreases by 2ai − 2. Wecall this a type I move. If k admits a type I move at Ai, we say k is stabilizedat Ai. If D and E are disjoint, we can slide the minimum above the maximumwith respect to h, decreasing w(k) by 4. This is called a type II move.

3. Cable knots and the companion torus

The knots we will study are called cable knots, a specific type of satelliteknot, defined below:

Definition 3.1. Suppose that K is a knot contained in a solid torus Vwith core C such that every meridian of V intersects K, and let J be anynontrivial knot in S3. Further, let ϕ : V → S3 be an embedding such thatϕ(C) is isotopic to J . Then we say that K = ϕ(K) is a “satellite knot”with companion J and pattern K. We call ∂ϕ(V ) the “companion torus”corresponding to J .

In general, if K is a satellite knot, K might not be isotopic into thecompanion torus. However, if its pattern K is a torus knot, we can pushthe pattern K into T = ∂ϕ(V ), an assumption that yields more informationthan the general case when we study the foliation of T induced by h. Thisleads to the definition of cable knots.


Figure 1: A (1, 5)-cable of the figure eight knot with the blackboardframing.

Definition 3.2. A “(p, q)-cable knot” is a satellite knot with pattern a(p, q)-torus knot K.

For an example of a cable knot, see figure 1. We follow the conventionof [7], where a (p, q)-torus knot in T has homology class p[μ] + q[λ] with μa meridian and λ a longitude of ϕ(V ). Observe that a (p, q)-torus knot hasintersection number q with any meridian of ϕ(V ).

As noted above, h induces a singular foliation, FS , of any surface S ⊂ S3

such that h|S is Morse. As in [15], we distinguish between the various typesof saddle points of FS :

Definition 3.3. Let c be a critical value corresponding to a saddle σ ofFS , so that some component of S ∩ h−1([c − ε, c + ε]) is a pair of pants P .Let s1 and s2 denote the components of ∂P lying in the same level. If eitherbounds a disk in S, we say that σ is an “inessential saddle.” If σ is notinessential, then it is an “essential saddle.”

In addition to type I and type II moves, we will require one more moveto systematically simplify cable knots: disk slides.

Definition 3.4. Suppose that some embedding k of a knot K is containedin a surface S such that h|S is Morse. Suppose further that D ⊂ S is a disksatisfying the following conditions:

546 Alexander Zupan

Figure 2: A disk slide.

1. ∂D is contained in some level surface h−1(r);

2. int(D) contains exactly one minimum or maximum;

3. some component of k ∩ D is an outermost arc γ that contains exactlyone critical point of h|k and cobounds a disk Δ with an arc in ∂D,where int(Δ) contains the critical point of D.

Then there is an isotopy through Δ supported in a neighborhood of Δ thattakes γ to an arc γ′ that cobounds a disk Δ′ with an arc in ∂D such thatint(Δ′) contains no critical points and int(Δ′) ∩ k = ∅. Additionally, γ′ canbe chosen so that it contains exactly one critical point occuring at the sameheight as the critical point of γ. We call the isotopy that replaces γ with γ′

a “disk slide,” pictured in figure 2.

Note that by requiring that the critical point of γ′ occur on the same levelas the critical point of γ, we ensure that if k′ is the result of performing a diskslide on k, we have w(k′) = w(k), although the width of k may temporarilyincrease during some intermediate step of the disk slide.

4. Efficient position of k

From this point forward, we set the convention that K is a (p, q)-cable knotwith companion J and companion torus T . As noted above, for any k ∈ K ,we may assume there is a torus isotopic to T containing k. In order to studyeach embedding k ∈ K with respect to various representatives in the isotopyclass of the companion torus T , we define the following collection:

K T = {(k, Tk) : k ∈ K , Tk ∼ T, k ⊂ Tk, and h|Tkis Morse}.

For an arbitrary element (k, Tk) of K T , the torus Tk is compressible onlyon one side, and we let Vk denote the solid torus bounded by Tk. We may alsoassume that the critical points of k and Tk occur at different levels. The goal


of this section is to show that any embedding k can be deformed to lie on atorus isotopic to T in an efficient way and without increasing w(k). To thisend, consider an arc γ of k embedded in an annulus A, where both bound-ary components of A are level and int(A) contains no critical points withrespect to h. Suppose that γ is an essential arc containing critical points,and let x denote the lowest maximum of γ. There are two points y, z ∈ γcorresponding to minima (one could be an endpoint) such that γ containsa monotone arc connecting x to y and x to z. Without loss of generality,suppose that h(y) > h(z). Then level arc components of A ∩ h−1(h(y) + ε)cobound disks D, E ⊂ A with arcs in γ such that ∂D contains exactly onemaximum (x), ∂E contains exactly one minimum (y) and int(D) ∩ γ =int(E) ∩ γ = ∅. Further, D ∩ E is a single point contained in γ. Refer tofigure 3.

If there is exactly one thick surface between x and y, then k admits atype I move canceling the two critical points. If there is more than one suchthick surface, we can slide x down along D, performing some number of typeII moves, until there is one such thick surface, after which k admits a type Imove. A similar argument is valid if γ is inessential in A or if γ is embeddedin a disk D with level boundary and one critical point, and γ contains morethan one critical point. We will use these ideas in the proof of the nextlemma, which is similar to the proof of Lemma 3.3 in [19] modified slightlyto accommodate the fact that for any (k, Tk) ∈ K T , we have k ⊂ Tk.

Lemma 4.1. For any k ∈ K , there exists (k′, Tk′) ∈ K T such that w(k′) ≤w(k) and the foliation of Tk′ by h contains no inessential saddles.

Figure 3: An essential arc γ containing critical points, shown with upperand lower disks D and E described above.

548 Alexander Zupan

Proof. Suppose that (k, Tk) ∈ K T and FTkcontains an inessential saddle.

By Lemma 1 (the Pop Over Lemma) of [15], we can modify Tk so that thereexists a saddle σ of FTk

with corresponding critical value c such that somecomponent P of Tk ∩ h−1([c − ε, c + ε]) is a pair of pants, with boundarycomponents s1, s2, s3 satisfying

1. s1 and s2 are contained in the same level surface L of h,

2. s1 bounds a disk D1 ⊂ Tk such that FTkrestricted to D1 contains only

one maximum or minimum,

3. D1 co-bounds a three-ball B with a disk D1 ⊂ L such that B does notcontain ±∞ and such that s2 lies outside of D1.

Without loss of generality, let D1 contain one maximum. By the above argu-ment, we can perform type I and II moves so that each component of k ∩ D1

contains exactly one maximum. Let A = P ∪ D1, so that A is an annuluswith boundary components s2 and s3. We can choose P so that k ∩ P con-sists only of vertical arcs. Then if k ∩ D1 �= ∅, k ∩ A contains inessentialarcs with exactly one critical point and boundary in s3. Let γ denote anoutermost arc in A, so that γ cobounds a disk Δ with an arc in s3 andint(Δ) ∩ k = ∅. If int(Δ) contains a critical point, we can perform a diskslide to remove it, after which can remove γ from k ∩ A, possibly by type IImoves.

Thus, after some combination of type I moves, type II moves and diskslides, we may assume that k ∩ D1 = ∅. By the proof of Lemma 3.2 in [19], wecan cancel the saddle σ with the maximum contained in D1 without increas-ing w(k). Repeating this process finitely many times finishes the proof. �

Hence, for the purpose of finding thin position we may assume that forany (k, Tk) ∈ K T , the foliation FTk

contains no inessential saddles, andso each disk D ⊂ Tk with level boundary must contain exactly one criticalpoint. We distinguish between two types of saddles:

Definition 4.1. Let (k, Tk) ∈ K T , with σ a saddle point of FTkand c

the corresponding critical value. Then some component P of T ∩ h−1([c − ε,c + ε]) is a pair of pants. We say that σ is a “lower saddle” if P ∩ h−1(c + ε)has two components; otherwise, σ is an “upper saddle.”

Next, for (k, Tk) ∈ K T , we suppose that the foliation of Tk containsonly essential saddles and decompose Tk into annuli as follows: for eachcritical value ci corresponding to a saddle, some component of Tk ∩ h−1


([ci − ε, ci + ε]) is a pair of pants, call it Pi. If Pi is a lower saddle, thenPi ∩ h−1(c − ε) is a simple closed curve that bounds a disk Di in S withexactly one minimum. If Pi is an upper saddle, there exists a similar diskDi with exactly one maximum. In either case, Ai = Pi ∪ Di is an annuluswhose foliation contains exactly one saddle and one minimum or maximum,called a lower annulus or an upper annulus, respectively.

Now Tk \ ∪Ai is a collection of vertical annuli whose foliations containno critical points, and we have decomposed S into a collection of lower,upper, and vertical annuli, which we denote {A1, . . . , Ar}. In addition, theser annuli as a collection have r distinct boundary components, which wedenote {δ1, . . . , δr}. Let L be any level two-sphere containing some δi. Of allcurves in Tk ∩ L that are essential in Tk, consider a curve α that is innermostin L. Thus α bounds a disk D in L containing no other essential simple closedcurves in Tk ∩ L. Potentially, D contains some inessential curves in Tk ∩ L,but these curves bound disks in Tk, so after some gluing operations, we seethat α bounds a compressing disk for Tk. Since Tk is compressible only onone side, it follows that α bounds a meridian disk of the solid torus Vk;hence, δi also bounds a meridian disk since it is parallel to α in Tk. AsK is a (p, q)-cable, each δi (oriented properly) has algebraic intersection qwith k, which implies k has at least q · r intersections with the collection{δ1, . . . , δr}.

We would like k to be contained in the companion torus as efficiently aspossible; for this purpose, we define efficient position, where we decomposeTk into annuli as describe above.

Definition 4.2. We say that (k, Tk) ∈ K T is an “efficient position” ifk ∩ Ai is a collection of essential arcs in Ai for every i. Further, if Ai is avertical annulus, we require that each arc of k ∩ Ai contain no critical points,and if Ai is an upper or lower annulus, we require that each arc of k ∩ Ai

contains exactly one minimum or maximum.

Note that if (k, Tk) is an efficition position, it is implicit in Definition 4.2that the foliation of Tk contains only essential saddles. Of course, given anarbitrary element (k, Tk) ∈ K T , we may not necessarily assume that (k, Tk)is an efficient position, but we may employ the next lemma.

Lemma 4.2. For any k ∈ K , there exists (k′, Tk′) ∈ K such that w(k′) ≤w(k) and (k′, Tk′) is an efficient position.

Proof. Let (k, Tk) ∈ K T such that FTkcontains no inessential saddles and

decompose Tk into annuli as described above. Suppose that∑ |k ∩ δi| is

550 Alexander Zupan

minimal up to isotopies that do not increase width. By previous arguments,we suppose that after a series of type I and type II moves, we have forevery vertical annulus Ai, k ∩ Ai consists of monotone essential arcs andinessential arcs with exactly one critical point, and for every lower or upperannulus Ai, k ∩ Ai consists of essential and inessential arcs with exactly onecritical point. As determined above,

∑ |k ∩ δi| ≥ q · r. If∑ |k ∩ δi| = q · r,

then every oriented intersection of k with δi must occur with the same sign,so every arc of k ∩ Ai is essential and (k, Tk) is an efficient position.

Conversely, if every arc of k ∩ Ai is essential for all i, then all orientedintersections of k with δi occur with the same sign and

∑ |k ∩ δi| = q · r.Thus, if

∑ |k ∩ δi| > q · r, there exists Aj such that k ∩ Aj contains aninessential arc. Let γ be an inessential arc that is outermost in Aj , so that γcobounds a disk Δ with a level arc in ∂Aj such that int(Δ) ∩ k = ∅. Possiblyafter a disk slide if Aj is upper or lower, we can slide γ along Δ to removetwo points of some k ∩ δi. This isotopy does not increase the width of k,contradicting the minimality assumption above. �

As a result of the proof of Lemma 4.2, if (k, Tk) is an efficient position,we have that k ∩ Ai consists of q essential arcs for each i. See figure 4.

5. The width of the companion torus and the width ofcable knots

Neglecting K for the moment, let L be any knot in S3 and L the set ofembeddings of S1 isotopic to L. In this section, we define the width of a torusin S3. However, instead of modifying the standard definition that counts thenumber of intersections of a knot with level two-spheres, we will keep trackof the order in which the upper and lower saddles occur in the foliationof the torus by h. In the case of knots, observe that we can calculate thewidth of any embedding l ∈ L if we know the order in which all minimaand maxima of l occur with respect to h. To formalize this notion, let Z bethe free monoid generated by two elements, m and M . Now, define Z to bethose elements σ = mα1Mβ1 . . . mαnMβn ∈ Z such that

1. αi, βi �= 0 for all i,

2.∑j

i=1 αi >∑j

i=1 βi for all j < n, and

3.∑n

i=1 αi =∑n

i=1 βi.

We define a map from L to Z, l → σl, by the following: Let c0 < · · · <cp be the critical values of h |l. We create a word σl consisting of p + 1


Figure 4: An example of the simplification suggested by Lemma 4.2. In thefirst picture, an inessential arc γ is contained in an upper annulus. After adisk slide, we remove γ from the upper annulus, and then remove it fromthe vertical annulus. Now the inessential arc contains three critical points,so after a type I move, we can remove it from the lower annulus, continuingas necessary until an efficient position is attained.

letters by mapping the tuple (c0, . . . , cp) to a word by assigning m to eachminimum and M to each maximum. Next, we define the width of an elementσ = mα1Mβ1 . . . mαnMβn ∈ Z by

w(σ) = 2(∑

αi

)2 − 4∑

i>j

αiβj .

The following lemma should be expected:

Lemma 5.1. For any l ∈ L , w(l) = w(σl).

Proof. Let σl = mα1Mβ1 . . . mαnMβn . Then the thick/thin tuple for l is(2α1, 2(α1 − β1), 2(α1 − β1 + α2), . . . , 2(α1 − β1 + α2 − · · · + αn)). From the

552 Alexander Zupan

width formula given by (2.1),

w(k) =12[(2α1)2 − (2(α1 − β1))2 + (2(α1 − β1 + α2))2

− · · · + (2(α1 − β1 + α2 − · · · + αn))2]

= 2[α21 + α2

2 + 2α2(α1 − β1)

+ · · · + α2n + 2αn(α1 − β1 + α2 − · · · − βn−1)]

= 2(∑

αi

)2 − 4∑

i>j

αiβj ,

as desired. �

As an example, consider the embedding l pictured in figure 5. A simpleverification shows that σl = m3MmM3 and w(l) = w(σl) = 28.

Next, we consider the collection of possible companion tori for a satelliteknot with companion L. Define SL to be the collection of embedded tori Ssuch that

1. h|S is Morse,

2. the foliation FS induced by h contains no inessential saddles, and

3. S is isotopic to the boundary of a regular neighborhood of L.

Now, as with L , we can define a map from SL to Z, S → σS , by the fol-lowing: Let c0 < · · · < cp be the critical values corresponding to the saddlesof FS . We create a word σS consisting of p + 1 letters by mapping the tuple

Figure 5: An embedding l with σl = m3MmM3.


(c0, . . . , cp) to a word by assigning m to each lower saddle and M to eachupper saddle. It is clear that σS ∈ Z, and slightly more difficult to see thatσS ∈ Z. The proof of this fact is left to the reader.

Using this association, we define the width of any torus S ∈ SL byw(S) = w(σS), and similar to the definition of knot width, we have

Definition 5.1. The “neighborhood width” of the knot L, nw(L), is given by

nw(L) = minS∈SL

w(S).

As a knot invariant, neighborhood width is not new; it is equivalent toknot width by the next lemma.

Lemma 5.2. For any knot L, nw(L) = w(L).

Proof. First, let l0 be an embedding of L such that w(l0) = w(L), and letS0 be the boundary of a regular neighborhood of l0 in S3 such that everysaddle of S0 occurs slightly above a minimum or slightly below a maximumof l0. All such saddles are easily seen to be essential, so S0 ∈ SL. Further,w(S0) = w(σS0) = w(σl0) = w(L); hence

nw(L) ≤ w(L).

For the reverse inequality, let S ∈ SL. Since S is isotopic to a regularneighborhood of L, we have that any longitude of S is isotopic to L. Decom-pose S into a collection of upper, lower, and vertical annuli {A1, . . . , Ar} asin Section 4. For each lower or upper annulus Ai, let li be an essential arcin Ai passing through the saddle with exactly one minimum or maximum.For each vertical annulus Ai, let li be a monotone arc connecting the twoendpoints of the arcs contained in the annuli adjacent to Ai (these annulimust contain saddles). Lastly, let l be the union of all the arcs li, so that lis a simple closed curve.

As shown above, the collection of boundary components {δi} of theannuli {Ai} bound meridian disks for the solid torus bounded by S, andsince l intersects each δi once, l is a longitude of S. By construction w(S) =w(σS) = w(σl) = w(l) ≥ w(L), and since this is true for all S ∈ SL, we have

nw(L) = w(L),

completing the proof. �

Finally, we have all the necessary tools to find the width of a cable knot.

554 Alexander Zupan

Theorem 5.1. Suppose that K is a (p, q)-cable of a nontrivial knot J . Then

w(K) = q2 · w(J).

Proof. Let j be an embedding of J such that w(j) = w(J), and let Tj be theboundary of a regular neighborhood of j such that every saddle of Tj occursslightly above a minimum or slightly below a maximum of j. Suppose thatσj = mα1Mβ1 . . . mαnMβn . We can cable j along Tj , creating an embeddingk ∈ K such that σk = mqα1M qβ1 . . . mqαnM qβn . By Lemma 5.1,

w(K) ≤ w(σk) = q2 · w(σj) = q2 · w(J).

We call k an “obvious” cabling of j.On the other hand, suppose k′ ∈ K is a given thin position. By

Lemmas 4.1 and 4.2, there exists a torus Tk′ such that (k′, Tk′) ∈ K T isan efficient position. Let σTk′ = mα′

1Mβ′1 . . . mα′

n′Mβ′n′ . Note that Tk′ ∈ SJ ,

and thus w(Tk′) ≥ w(J) by Lemma 5.2. We decompose Tk′ into a collec-tion of upper, lower, and vertical annuli {A1, . . . , Ar} as above in Sec-tion 4. For each upper annulus Ai, k′ ∩ Ai consists of q essential arcs,each containing one maximum. Suppose ci is the critical value correspond-ing to the saddle in Ai. Then there is an isotopy of k supported in Ai

taking the q arcs to arcs in h−1((ci, ci + ε]), and this isotopy does notincrease w(k′). A similar statement is true for each lower annulus Ai. As allcritical points of k′ are contained in upper or lower annuli, we can computeσk′ = mqα′

1M qβ′1 . . . mqα′

n′M qβ′n′ , and thus

w(K) = w(mqα′1M qβ′

1 . . . mqα′n′M qβ′

n′ ) = q2 · w(Tk′) ≥ q2 · w(J),

completing the proof. �

The proof of the theorem reveals that if k is an “obvious” cabling of athin position of J , then it is a thin position of K.

Corollary 5.1. For a knot J in S3, the following are equivalent:

(a) J is bridge-thin.

(b) Every cable of J is bridge-thin.

(c) There exists a cable of J that is bridge-thin.


6. Meridional smallness of cable knots

In the previous section, we showed that bridge-thinness is a property pre-served by cabling. Here we wish to show the same is true for meridionalsmallness and mp-smallness, defined below. We need several other defini-tions first (we assume the reader is familiar with the definitions of incom-pressibility and ∂-incompressibility).

Definition 6.1. A surface S ⊂ M is “essential” if it is incompressible,∂-incompressible and not parallel to ∂M .

Definition 6.2. A knot L ⊂ S3 is “meridionally small” if E(L) containsno essential surface S with ∂S consisting of meridian curves of η(K), whereη denotes a regular neighborhood. A knot L ⊂ S3 is “meridionally planarsmall,” or mp-small, if E(L) contains no essential planar surface S withmeridional boundary.

Note that if S is a properly embedded surface with meridional boundaryin a knot exterior E(L) and S is ∂-compressible, then either S is compressibleor ∂-parallel. In addition, all components of ∂E(L) \ η(∂S) are annuli, so ifS is ∂-parallel, S must be an annulus. Thus to show S is essential it sufficesto show S is incompressible and not a ∂-parallel annulus.

Recall that K is a (p, q)-cable of J , with pattern K a (p, q)-torus knotcontained in a solid torus V . Letting Cp,q = V \ η(K), we observe that we candecompose E(K) as E(J) ∪ Cp,q, where the attaching map depends on theframing ϕ : V → S3 that maps a core of V to J . Following [7], we call Cp,q a(p, q)-cable space. See figure 6. Note that Cp,q has a Seifert fibering with oneexceptional fiber (a core of V ), and it has two torus boundary components,∂V (the outer boundary, denoted ∂+Cp,q) and ∂ η(K) (the inner boundary,denoted ∂−Cp,q). In order to understand essential meridional surfaces inE(K), we require the next lemma. We note that this lemma is implied byLemma 3.1 of [7], but we do not need its full generality and so we providean elementary proof. For a treatment of Seifert fibered spaces, see [8]. Asurface in a Seifert fibered space is horizontal if it is transverse to all fibersand vertical if it is a union of fibers.

Lemma 6.1. Suppose S is a connected incompressible surface in Cp,q,where S ∩ ∂−Cp,q �= ∅ and each component of S ∩ ∂−Cp,q is a meridian curve.Then S is a disk with q punctures, and S ∩ ∂+Cp,q is a meridian curve of V .

556 Alexander Zupan

Figure 6: The cable space C3,2.

Proof. Since Cp,q is Seifert fibered, every incompressible surface is either hor-izontal or vertical by Hatcher [8]. By assumption S has meridional boundaryin ∂−Cp,q, so S is horizontal. We can extend S to a horizontal surface S′ in aSeifert fibered solid torus by attaching meridian disks to each component ofS ∩ ∂−Cp,q and gluing η(K) back into Cp,q. But every connected horizontalsurface in such a torus is a meridian disk, so S′ is a meridian disk, fromwhich it follows that S is a disk with q punctures and ∂S ∩ ∂+Cp,q = ∂S′ isa meridian curve. �

On the other hand, let S′ be a meridian disk of V intersecting K mini-mally, with S = S′ ∩ Cp,q. Then Cp,q cut along S is homeomorphic to S × I,from which it follows that S is incompressible in Cp,q. We use these facts inthe proof of the next theorem.

Theorem 6.1. For a knot J in S3, the following are equivalent:

(a) J is meridionally small.

(b) Every cable of J is meridionally small.

(c) There exists a cable of J that is meridionally small.

Proof. First we prove that (c) implies (a) or, equivalently, if J is not merid-ionally small and K is a cable of J , then K is not meridionally small. If Jis not meridionally small then E(J) contains an essential meridional surface


S, and let K be a (p, q)-cable of J , so that E(K) = E(J) ∪T Cp,q, whereT = ∂E(J). Then every component of ∂S bounds a disk with q puncturesin Cp,q, and thus we can construct a meridional surface S′ in E(K) by glu-ing such a disk D to each boundary component of ∂S. We claim that S′ isessential in E(K). By the above, S′ ∩ Cp,q is incompressible in Cp,q. Notethat |∂S′| = q · |∂S| > 2, so S′ is not a ∂-parallel annulus. Suppose Δ is acompressing disk for S′ in E(K). If Δ ∩ T = ∅, then either Δ ⊂ Cp,q, whichis ruled out by the argument above, or Δ ⊂ E(J), which contradicts theincompressibility of S.

Thus, Δ ∩ T �= ∅, and suppose Δ is chosen so that |Δ ∩ T | is minimal.Since T is incompressible in E(K), Δ ∩ T contains no simple closed curves.Let α be an arc in Δ ∩ T that is outermost in Δ, so that α cobounds a diskΔ′ ⊂ Δ with an arc β ⊂ ∂Δ such that int(Δ′) ∩ T = ∅. There are two casesto consider: First, suppose that Δ′ ⊂ Cp,q. Then both endpoints of α arecontained in the same component of S′ ∩ T , which means that α is inessentialin an annular component of T \ S′ and cobounds a disk Δ′′ ⊂ T with an arcγ ⊂ S′ ∩ T . Gluing Δ′ to Δ′′ along α yields a disk Δ∗ with ∂Δ∗ ⊂ S′ ∩ Cp,q,and by the incompressibility of S′ ∩ Cp,q in Cp,q, ∂Δ∗ = β ∪ γ bounds a diskD contained in S′ ∩ Cp,q. Sliding β along D, we can remove at least oneintersection of Δ with T , contradicting the minimality of |Δ ∩ T |.

In the second case, suppose that Δ′ ⊂ E(K). If α is an inessential arc insome annular component of T , the above argument holds. If α is essential,then Δ′ is a ∂-compressing disk for S in E(J), contradicting the assumptionthat S is essential. We conclude that S′ is an essential meridional surface inE(K), showing that (c) implies (a).

Now we show that (a) implies (b) or, equivalently, if some cable of J isnot meridionally small then J is not meridionally small. Suppose that thereexists a (p, q)-cable K of J such that E(K) contains an essential meridionalsurface R. As above, E(K) = E(J) ∪T Cp,q. Assume |R ∩ T | is minimal up toisotopy. We claim that R ∩ Cp,q is incompressible in Cp,q. Suppose not. Thenthere is a compressing disk Δ ⊂ Cp,q for R ∩ Cp,q. Since R is incompressiblein E(K), ∂Δ bounds a disk Δ′ ⊂ R, and by isotopy we can replace Δ′ withΔ and reduce the number of intersections of R with T , a contradiction. ByLemma 6.1, each component of R ∩ Cp,q is a punctured disk, and it followsthat R′ = R ∩ E(J) is a meridional surface. If R′ is a ∂-parallel annulusin E(J), we can lower |R ∩ T | via isotopy, contradicting the minimality of|R ∩ T |. Likewise, if R′ is compressible in E(J), we can lower |R ∩ T | asin the case of R ∩ Cp,q above. Thus, R′ is an essential meridional surfacein E(J), and (a) implies (b). Clearly, (b) implies (c), completing the proof.

�

558 Alexander Zupan

It should be noted that in the above proof, if either surface S or R isplanar, then the induced surface S′ or R′ is also planar, yielding:

Theorem 6.2. For a knot J in S3, the following are equivalent:

(a) J is mp-small.

(b) Every cable of J is mp-small.

(c) There exists a cable of J that is mp-small.

7. Destabilizing non-minimal bridge positions of cables

If l is a bridge position for a knot L, then l has exactly one thick levelA = h−1(a1). We call this level a bridge sphere and say that two bridgepositions l and l′ equivalent if there exists an isotopy ft, called a bridgeisotopy, taking l to l′ such that for all t, ft(A) is a bridge sphere for ft(l). Ifl is a bridge position admitting a type I move, we say that l is stabilized. Inthis context a type I move is also called a destabilization. Since a type I movecancels a minimum and a maximum, it is clear that every minimal bridgeposition is not stabilized. For K a (p, q)-cable of J , we have the following:

Lemma 7.1. Suppose that k is a bridge position. Then there exists (k′, Tk′)∈K T such that k′ is equivalent to k and (k′, Tk′) is an efficient position, ork is stabilized.

Proof. Let (k, Tk) ∈ K T . By Lemma 2.4 of [12], there exists (k′, Tk′) ∈K T such that k′ is equivalent to k and all saddles in the foliation of Tk′ byh are essential. Note that any disk slide of k′ is supported in a neighborhoodaway from a bridge sphere; thus, if k′′ is the result of a disk slide on k′,k′′ is equivalent to k′. Now, as in Section 4, we split Tk′ into a collection{A1, . . . , Ar} of vertical, upper and lower annuli, where each component ofk′ ∩ Ai is an arc containing at most one critical point, or else a minimum andmaximum of some arc can be canceled and k′ is stabilized. If Ai is verticaland some arc component α of k′ ∩ Ai is inessential, outermost in Ai, andcontains a maximum, we can push α off Ai and onto the adjacent lowerannulus Aj , where its maximum cancels a minimum of an arc componentof k′ ∩ Aj . A similar argument holds if α contains a minimum or if α isan inessential arc in an upper or lower annulus, although in this case wemay require a disk slide before pushing α. See figure 4. We conclude either(k′, Tk′) is an efficient position or k′ admits a destabilization. �


In [12], Makoto Ozawa shows that if K is a torus knot, then every non-minimal bridge position of K is stabilized. We essentially follow his proof,which we will summarize here, to show that if J is mp-small and every non-minimal bridge position of J is stabilized, then every cable K of J has thesame property. Ozawa utilizes the following theorem, proved by Hayashi andShimokawa [9] and later by Tomova [18]:

Theorem 7.1. If k ∈ K is a bridge position admitting a type II move,then either k is stabilized or E(K) contains an essential meridional planarsurface.

Thus, if K is mp-small, any bridge position admitting a type II move isstabilized.

From this point on, suppose J is mp-small. By Theorem 6.2, K is alsomp-small. We recall that for (k, Tk) ∈ K T , the solid torus bounded by Tk

is denoted Vk. Although Ozawa [12] deals specifically with torus knots, someresults apply in the setting of mp-small cable knots. Lemmas 3.2 to 3.4 of [12]together imply

Lemma 7.2. Suppose k is a bridge position. Then there exists (k′, Tk′) ∈K T an efficient position such that k′ is equivalent to k and a bridge sphereA = h−1(a1) such that all upper saddles of Tk′ occur above A, all lowersaddles occur below A, and Vk′ ∩ A is a collection of disks, or k is stabilized.

Thus, h foliates the solid torus Vk′ by disks, and any such (k′, Tk′) inducesa bridge position j of J by taking j a longitude of Tk′ as constructed inLemma 5.1. This fact also ensures that any upper or lower disk for j can bechosen to miss int(Vk′) and thus can easily be modified to an upper or lowerdisk for k′. Finally, we have

Theorem 7.2. Suppose K is a (p, q)-cable with companion J , where J ismp-small.

(a) If every non-minimal bridge position of J is stabilized, then every non-minimal bridge position of K is stabilized.

(b) The cardinality of the collection of minimal bridge positions of K doesnot exceed the cardinality of the collection of minimal bridge positionsof J .

Proof. (a) Suppose k ∈ K is a bridge position. By Lemmas 7.1 and 7.2, wemay pass to an equivalent bridge position and assume there exists (k, Tk) ∈

560 Alexander Zupan

K T an efficient position such that h foliates Vk by disks, or else k is sta-bilized. Let A be a bridge sphere for k guaranteed by Lemma 7.2, and notethat the induced bridge position j of J satisfies b(j) = 1

2 |A ∩ Tk|. There aretwo cases to consider: First, suppose that |A ∩ Tk| = 2 · b(J). Since eachupper annulus contains q maxima, we have b(k) = q · b(J). By [14, 15],b(K) = q · b(J), and thus k is a minimal bridge position of K. On the otherhand, suppose that |A ∩ Tk| > 2 · b(J). Then j is a non-minimal bridge posi-tion for J , and by assumption j is stabilized. It follows that k is stabilized,finishing the first part of the proof.

(b) For each minimal bridge position j of J , assign an obvious cablingk ⊂ ∂η(j) of j with q · b(J) maxima. It suffices to prove that the associationϕ : j → k is surjective. First, suppose that k′ is another obvious cabling ofa bridge position of j′ equivalent to j with q · b(J) maxima. Then there is abridge isotopy taking j′ to j, which induces an isotopy from ∂η(j′) to ∂η(j)which is also a bridge isotopy of k′. Since k is isotopic to k′ in ∂η(j), we canremove intersections of k and k′ and then push k′ onto k by a bridge isotopy,and so ϕ is well-defined. Now, if k is a minimal bridge position of K, wemay pass to an equivalent bridge position and assume there exists (k, Tk)such that h foliates Vk by disks by Lemma 7.2. Thus, the induced minimalbridge position j for J from Lemma 5.1 satisfies ϕ(j) = k, as desired. �

From [11, 12], we have

Corollary 7.1. If K is an n-fold cable of a torus knot or a two-bridge knot,then any non-minimal bridge position of K is stabilized. Additionally, if Kis an n-fold cable of a torus knot, it has a unique minimal bridge position,and if K is an n-fold cable of a two-bridge knot, it has at most two minimalbridge positions.

In [3], Coward proves that if J is a hyperbolic knot, then J has finitelymany minimal bridge positions. Hence

Corollary 7.2. If K is a (p, q)-cable of J , where J is hyperbolic and mp-small, then K has finitely many minimal bridge positions.

References

[1] R. Blair, Bridge number and Conway products, Algebr. Geom. Top. 10(2010), 789–823.

[2] R. Blair and M. Tomova, Width is not additive, arXiv:1005.1359.


[3] A. Coward, Algorithmically detecting the bridge number of hyperbolicknots, arXiv:0710.1262.

[4] M. Culler, C. McA. Gordon, J. Luecke and P.B. Shalen, Dehn surgeryon knots, Ann. Math. 125 (1987), 237–300.

[5] D. Gabai, Foliations and the topology of 3-manifolds III, J. DifferentialGeom. 26 (1987), 479–536.

[6] D.J. Heath and T. Kobayashi, Essential tangle decomposition from thinposition of a link, Pacific J. Math. 179 (1997), 101–117.

[7] C. McA. Gordon and R.A. Litherland, Incompressible planar surfacesin 3-manifolds, Topol. Appl. 18(2–3) (1984), 121–144.

[8] A. Hatcher, Notes on basic 3-manifold topology, unpublished.

[9] C. Hayashi and K. Shimokawa, Thin position of a pair (3-manifold,1-submanifold), Pacic J. Math. 197 (2001), 301–324.

[10] K. Morimoto, Tunnel number, connected sum and meridional essentialsurfaces, Topology 39 (2000), 469–485.

[11] J.-P. Otal, Presentations en ponts des noeuds trivial, C. R. Acad. Sci.Paris Ser. I Math. 294 (1982), 553–556.

[12] M. Ozawa, Non-minimal bridge positions of torus knots are stabilized,Math. Proc. Cambridge Philos. Soc. 151 (2011), 307–317.

[13] M. Scharlemann and J. Schultens, 3-manifolds with planar presenta-tions and the width of satellite knots, Trans. Amer. Math. Soc. 358(2006), 3781–3805.

[14] H. Schubert, Uber eine numerische Knoteninvariante, Math. Z. 61(1954), 245–288.

[15] J. Schultens, Additivity of bridge numbers of knots, Math. Proc. Cam-bridge Philos. Soc. 135 (2003), 539–544.

[16] J. Schultens, Width complexes for knots and 3-manifolds, Pacific J.Math. 239(1) (2009), 135–156.

[17] A. Thompson, Thin position and bridge number for knots in the3-sphere, Topology 36(2) (1997), 505–507.

[18] M. Tomova, Thin position for knots in a 3-manifold, J. Lond. Math.Soc. (2) 80(1) (2009), 85–98.

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[19] A. Zupan, A lower bound on the width of satellite knots, Top. Proc. 40(2012), 179–188.

14 Maclean HallDepartment of MathematicsUniversity of IowaIowa CityIA 52242USAE-mail address: [email protected]

Received October 19, 2010

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